Log-normal distribution from a process that is not multiplicative but is additive

Area of Science:

• Probability Theory
• Statistical Mechanics
• Stochastic Processes

Background:

• The central limit theorem (CLT) states that sums of independent random variables approach a Gaussian distribution as the number of variables increases.
• Log-normal distributions are typically associated with multiplicative processes or nonlinear systems.
• Current understanding posits that additive processes lead to Gaussian distributions.

Purpose of the Study:

• To investigate the limiting distribution of sums of positive random variables.
• To challenge the established link between log-normal distributions and multiplicative processes.
• To demonstrate that additive processes can also result in log-normal distributions.

Main Methods:

• Analysis of sums of a specific class of positive random variables.
• Derivation of conditions under which sums exhibit log-normal behavior.
• Application of findings to random scalar fields, including turbulence.

Main Results:

• Sums of positive random variables exhibit a faster convergence to a log-normal distribution than to a Gaussian distribution.
• The log-normal distribution is observed for sums (additive processes) under certain conditions.
• This contrasts with the prevailing view that log-normal distributions arise solely from products (multiplicative processes).

Conclusions:

• Log-normal distributions are not exclusive to multiplicative processes and can emerge from additive processes.
• The study reveals conditions for observing log-normal distributions in sums of random variables.
• The findings have implications for understanding phenomena like random scalar fields and turbulence.